From what I understand, a photon of "sufficient" energy will interact with a free electron such that the recoil electron receives part of the energy of the photon, and the scattered photon has a Doppler shift (change in wavelength). If the scattered photon still has "sufficient" energy, Compton scattering of that photon may be repeated. What if the scattered photon has "insufficient" energy? Is there any effect at all? Can a photon's energy be totally dissipated by a series of photon-electron interactions?